August 7 (Wednesday)
13h30 - 00h00 Registration
14h20 - 14h30 Opening Ceremony
14h30 - 18h00 Session 7A
14h30 - 15h30 7A-1 Keynote Lecture I / slide
Speaker Richard P. Stanley, MIT, USA
Title Two enumerative tidbits
Abstract We discuss two unrelated results in enumerative combinatorics.
(i) Smith normal form of a matrix related to Young diagrams. We generalize a result of Carlitz, Roselle, and Scoville on a combinatorial matrix of determinant one by introducing additional parameters and computing the Smith normal form of the resulting matrix.
(ii) A distributive lattice associated with three-term arithmetic progressions (with Fu Liu). We prove two conjectures of Noam Elkies related to arithmetic progressions of length three by showing a connection with a distributive lattice of certain semistandard Young tableaux.
15h45 - 16h45 7A-2 Talk I
Speaker Richard A. Askey, University of Wisconsin-Madison, USA
Title The magical connection between balanced and well-poised series, both hypergeometric and basic hypergeometric
Abstract Hypergeometric series with the q-binomial theorem. There are two natural first steps one can take.
(1-x)a(1-x)b = (1-x)a+b
(1-x)a(1+x)a = (1-x2)a
When the binomial theorem is used on each series and the coefficients of xn are equated, the first identity is an instance of what will become balanced series when more parameters are introduced, and the second is an instance of well-poised series, and later very well-poised series. At this stage the two identities have nothing to do with each other. At a higher level the two chains of identities become related, Whipple's formula for hypergeometric series and Watson's extension for basic hypergeometric series. Other connections will be mentioned, and two different extensions beyond this level will be described. One, which is due to George Andrews, contains variants of the Rogers-Ramanujan identities. The other has a mysterious twist which I do not understand.
17h00 - 18h00 7A-3 Talk II / slide
Speaker Sangwook Kim, Chonnam National University, Korea
Title Enumeration of Schröder families by type
Abstract Schröder paths, sparse noncrossing partitions, and partial horizontal strips are three classes of Schröder objects which carry a notion of type. We provide type-preserving bijections among these objects and an explicit formula which enumerates these objects according to type and length. We also define a notion of connectivity for these objects and discuss an analogous formula which counts connected objects by type. This is joint work with Suhyung An and Sen-Peng Eu.
August 8 (Thursday)
10h00 - 12h30 Session 8A
10h00 - 11h00 8A-1 Talk III / slide
Speaker Mitsugu Hirasaka, Pusan National University, Korea
Title Zeta functions of adjacency algebras induced by graphs
Abstract Let Γ denote a finite digraph and RΓ denote the subring generated by the adjacecny matrix of Γ In this talk we focus on the number an of ideals of RΓ with index n, and show a way to find the formal Dirichlet series
Σn≥1 an n-s
when Γ can be a relation of an association scheme of prime order. This is a joint work with Akihide Hanaki.
11h30 - 12h30 8A-2 Talk IV / slide
Speaker Jiang Zeng, Université Claude Bernard Lyon 1, France
Title Jacobi-Stirling numbers and Jacobi-Stirling permutations
Abstract The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second order Jacobi differential expression. They are refinements of the Legendre-Stirling numbers and generalize the Stirling numbers and central factorial numbers. In this talk I will report on the recent work about the combinatorics of these numbers. Moreover, the diagonal generating functions of Jacobi-Stirling numbers are rational functions, of which the numerators are the enumerative polynomials of the Jacobi-Stirling permutations.
12h30 - 12h40 Group Photo
12h40 - 14h30 Lunch
14h30 - 18h00 Session 8B
14h30 - 15h30 8B-1 Keynote Lecture II / slide
Speaker Richard P. Stanley, MIT, USA
Title Polynomial sequences of binomial type
Abstract A sequence p0(n), p1(n), ... of complex polynomials is of binomial type if p0(n)=1 and
Σk≥0 pk(n) xk / k! = ( Σk≥0 pk(1) xk / k! )n.
Such polynomials play a fundamental role in the theory of operator calculus developed by G.-C. Rota and his collaborators. After briefly reviewing this theory, we will focus on examples of such polynomials. In particular, we discuss recent work of J. Schneider related to placing figures on tori. An application is given to chromatic polynomials of toroidal grid graphs.
15h45 - 16h45 8B-2 Talk V / slide
Speaker Mourad E. H. Ismail, University of Central Florida, USA
Title Combinatorics of 2D-Hermite polynomials
Abstract We discuss the combinatorics and generating functions of the 2D-Hermite polynomials. This is partly based on joint work with Plamen Simeonov.
17h00 - 18h00 8B-3 Talk VI / slide
Speaker Suyoung Choi, Ajou University, Korea
Title Introduction to the a-number of graphs and hypergraphs
Abstract Recently, I and my colleague Park have introduced new combinatorial invariants, called the a-number, of any finite simple graph, which arise in toric topology. Interestingly, for specific families of the graph, our invariants are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers. In this talk, I introduce several further works on this topic, and I will discuss about the analogue of the invariant for hypergraphs.
18h00 - 21h00 Banquet
August 9 (Friday)
10h00 - 12h30 Session 9A
10h00 - 11h00 9A-1 Talk VII / slide
Speaker Soojin Cho, Ajou University, Korea
Title Littlewood-Richardson numbers of Schur's S- and P-Functions
Abstract Littlewood-Richardson numbers(LR-numbers) are structure constants of Schur's S-functions. Many combinatorial models for LR-numbers of Schur functions are known and they are very well understood. We review known combinatorial rules to calculate LR-numbers and interesting properties of LR-numbers including symmetries and factorization theorem. We then introduce eight useful reduction formulae deleting one or two rows (columns) of each partition. As an application, we prove that if the LR-number is 1 and each partition has distinct parts, then one of two types of our reduction formulae is always applicable and hence we have an algorithm to test if the LR-number is 1.
There do not exist so many combinatorial models of LR-numbers for Schur's P-functions. We introduce a new LR-rule for Schur's P-functions and some properties of them.
11h30 - 12h30 9A-2 Talk VIII / slide
Speaker Sen-Peng Eu, National University of Kaohsiung, Taiwan
Title Signed Counting of Euler numbers
Abstract Euler numbers count several important classes of permutations, among them the alternating permutations and the simsun permutations. In this talk we introduce some new results on the signed counting of these permutations.
12h30 - 14h30 Lunch
14h30 - 18h00 Session 9B
14h30 - 15h30 9B-1 Keynote Lecture III / slide
Speaker Richard P. Stanley, MIT, USA
Title Valid orderings of hyperplane arrangements
Abstract Let A be a finite set of hyperplanes in Rn. Let L be a sufficiently generic directed line in Rn Then L intersects the hyperplanes in A in a certain order, called a valid ordering of A. We will discuss connections between valid orderings and such topics as line shellings of polytopes, the Dilworth truncation of a matroid, and a generalization of chromatic polynomials. Some knowledge of matroid theory will be useful but not essential for understanding this lecture.
15h45 - 16h45 9B-2 Talk IX / slide
Speaker Dennis Stanton, University of Minnesota, USA
Title Reflection factorizations of Singer cycles
Abstract The number of factorizations of an n-cycle in Sn into n-1 transpositions is nn-2. We consider a version of this theorem when GLn(Fq) replaces Sn, the Singer cycle replaces an n-cycle, and reflections replace transpositions. We give explicit enumeration formulas for this question, and also longer factorizations. The answers involve a mixture of binomial and q-binomial coefficients. Character techniques are used, no bijective proofs are known. This is joint work with Joel Lewis and Vic Reiner.
17h00 - 18h00 9B-3 Talk X / slide
Speaker Jang Soo Kim, KIAS, Korea
Title Moments of Askey-Wilson polynomials
Abstract The Askey-Wilson polynomials are the most general orthogonal polynomials among those classified by the Askey scheme. These are orthogonal polynomials in one variable with 5 parameters. In this talk I will talk about 3 combinatorial methods to study the n-th moment of the Askey-Wilson polynomials. The first method is Viennot's theory of weighted Motzkin paths. The second method uses staircase tableaux introduced by Corteel and Williams. The third method is a modification of an idea of Ismail, Stanton, and Viennot on matchings and q-Hermite polynomials. Using the third method we express the n-th moment as a fraction of two generating functions for certain matchings and obtain a new formula for the moment. This is joint work with Dennis Stanton.
18h00 - 00h00 Closing Ceremony